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Theorem lejoin1 17617
Description: A join's first argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)
Hypotheses
Ref Expression
joinval2.b 𝐵 = (Base‘𝐾)
joinval2.l = (le‘𝐾)
joinval2.j = (join‘𝐾)
joinval2.k (𝜑𝐾𝑉)
joinval2.x (𝜑𝑋𝐵)
joinval2.y (𝜑𝑌𝐵)
joinlem.e (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
Assertion
Ref Expression
lejoin1 (𝜑𝑋 (𝑋 𝑌))

Proof of Theorem lejoin1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 joinval2.b . . 3 𝐵 = (Base‘𝐾)
2 joinval2.l . . 3 = (le‘𝐾)
3 joinval2.j . . 3 = (join‘𝐾)
4 joinval2.k . . 3 (𝜑𝐾𝑉)
5 joinval2.x . . 3 (𝜑𝑋𝐵)
6 joinval2.y . . 3 (𝜑𝑌𝐵)
7 joinlem.e . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
81, 2, 3, 4, 5, 6, 7joinlem 17616 . 2 (𝜑 → ((𝑋 (𝑋 𝑌) ∧ 𝑌 (𝑋 𝑌)) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧)))
98simplld 767 1 (𝜑𝑋 (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  cop 4534   class class class wbr 5033  dom cdm 5523  cfv 6328  (class class class)co 7139  Basecbs 16478  lecple 16567  joincjn 17549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-lub 17579  df-join 17581
This theorem is referenced by:  joinle  17619  latlej1  17665
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